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Suppose you have a family of polynomials $$P_n(x)=\sum_{k=0}^n(-1)^ka_k^{(n)}x^k$$ for $n=0,1,2,\dots$.

Further assumptions: (1) the coefficients $a_k^{(n)}$ are recursively related to the $a_j^{(n-1)}$;

(2) the roots of each $P_n(x)$ are distinct positive real roots.

QUESTION. Are there techniques to generate an asymptotic (1st order is fine) for the largest root $\lambda_n$ of $P_n(x)$ as $n\rightarrow\infty$?

To be a bit more concrete on the "recurrence": say, $$a_k^{(n)}=f(k)a_k^{(n-1)}+g(n,k)a_{k-1}^{(n-1)}.$$ I hope this helps for being specific.

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    $\begingroup$ Depends on how the coefficients are related, but sometimes, one can find the root distribution in the limit, and thus the largest root, ncbi.nlm.nih.gov/pmc/articles/PMC388687 $\endgroup$
    – Per Alexandersson
    Commented Nov 10, 2016 at 0:26
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    $\begingroup$ Can you elaborate on your statement $a_k^{(n)}$ are recursively related to the $a_j^{(n-1)}$? There are infinitely many possible recursive relations and you cannot expect to deal with all of them in the same fashion. $\endgroup$
    – Liviu Nicolaescu
    Commented Nov 10, 2016 at 0:28
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    $\begingroup$ @Per Alexanderson: largest root is a more subtle thing then the "root distributon". By root distribution they usually understand a probablity measure (what draction of the roots lie on an interval) and it is not sensitive to a single root. $\endgroup$
    – Alexandre Eremenko
    Commented Nov 10, 2016 at 0:42
  • $\begingroup$ Very much depends on the exact nature of your "recurrence relation". Be more specific. $\endgroup$
    – Alexandre Eremenko
    Commented Nov 10, 2016 at 0:43

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